Originally motivated by a problem of Erdős on the maximum number of maximum-area triangles determined by an n-point set, we will discuss some new and old results on the following family of problems: Given a set of n points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact by sharing and/or interleaving vertices. We will present old and new results and techniques and open problems.